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'Into the Exponential Distribution' printed from http://nrich.maths.org/
This chart show the probability density functions for two
exponential distributions. Here are some questions to consider
concerning these sorts of distributions. You can use a mixture of
probability-based reasoning and algebraic reasoning based on the
properties of integrals and areas.
1. Which exponential distributions do these curves correspond
to? What interesting mathematical properties do they have? Which
has the largest mean?
2. Two separate areas are enclosed by the red and blue curves:
one to the left of the point of intersection and one to the right.
What can you say about the sizes of these
two areas? Can you give a good explanation?
3. Can you find the point of intersection of the pdfs? Can you
find the area of each of the enclosed parts
of the diagrams?
4. From the graphs, I visually estimated that $P(0.5 < Red
< 0.7) = 0.125$. Can you work out how I made this estimate? Is
it an over-estimate or an underestimate? How close do you think
that estimate is to the correct answer? Work out the correct answer
to find out.
5. Try to make, assess and check your own estimates similar to
that in question 4.